Effective approximation of heat flow evolution of the Riemann $$\xi $$ function, and a new upper bound for the de Bruijn–Newman constant

The Riemann hypothesis is true up to 3·1012

“…The next entry in [26, Table 1] is conditional on taking a little higher than , which of course, is not achieved by Theorem 1. This would enable one to prove .…”

“…We note that we can make an instant, but very mild, improvement on this. The second row in [26, Table 1, p. 65] shows that one may take provided one has shown . This leads to the following.…”

A Dynamical Systems Framework for Generating the Riemann Zeta Function and Dirichlet L-functions

“…The proposed approach could also be used to explain the dynamics of the Riemann Xi function zeros observed in other dynamical systems approaches. For example in the approach proposed in (Bruijn, 1950), the dynamics of the zeros was quantified using a de Bruijn-Newman constant (Newman, 1976) and subsequent results (Ki, 2009;Polymath, 2019;Newman, 2019) focused on improving the bounds of this constant towards zero. Using equation 83 it can be seen that for σ → 0 − and σ 2 = 1 − ω 2 , the LHS converges implying that the zeros of the Riemann Xi function should line up on the s = 1 2 + jω axis.…”

Orthogonal polynomial expansions for the Riemann xi function in the Hermite, Meixner--Pollaczek, and continuous Hahn bases

“…The lower bound was pushed upwards further in a succession of papers [18,19,20,40,41,60,58], with the bounds established gradually growing extremely close to 0 on the negative side. Most recently Rodgers and Tao [59] succeeded in proving Newman's conjecture that Λ ≥ 0, and recent work by the Polymath15 project [54] strengthened the result of Ki, Kim and Lee mentioned above by proving the sharper upper bound Λ ≤ 0.22. We now come to a key idea that relates the above discussion to our theme of expansions of the Riemann xi function in families of orthogonal polynomials, and the Hermite polynomials in particular.…”

“…That is to say, Pólya's theorem states that if an entire function G(z) is expressed as the Fourier transform of a function F (x) of a real variable, and all the zeros of G(z) are real, then, under certain assumptions of rapid decay on F (x) (see [10] for details), the zeros of the Fourier transform of F (x)e λx 2 are also all real. This discovery spurred much follow-up work by de Bruijn [10], Newman [38] and others [17,18,19,20,31,40,41,54,59,60,58] on the subject of what came to be referred to as the de Bruijn-Newman constant; the rough idea is to launch an attack on RH by generalizing the Fourier transform (1.11) through the addition of the "universal factor" e λx 2 inside the integral, and to study the set of real λ's for which the resulting entire function has only real zeros. See Section 2.5, where some additional details are discussed, and see [9,Ch.…”